## Abstract

We analyse electron and hole transport in organic light-emitting diodes (OLEDs) via the drift–diffusion equations. We focus on space-charge-limited transport, in which rapid variations in charge carrier density occur near the injecting electrodes, and in which the electric field is highly non-uniform. This motivates our application of singular asymptotic analysis to the drift–diffusion equations. In the absence of electron–hole recombination, our analysis reveals three regions within the OLED: (i) ‘space-charge layers’ near each electrode whose width is much smaller than the device width , wherein carrier densities decay rapidly and the electric field is intense; (ii) a ‘bulk’ region whose width is on the scale of , where carrier densities are small; and (iii) intermediate regions bridging (i) and (ii). Our analysis shows that the current scales as , where is the applied voltage, is the permittivity and is the electric mobility, in contrast to the familiar diffusion-free scaling . Thus, diffusion is seen to lead to a large increase in current. Finally, we derive an asymptotic recombination–voltage relation for a kinetically limited OLED, in which charge recombination occurs on a much longer time scale than diffusion and drift.

## 1. Introduction

Organic light-emitting diodes (OLEDs) are solid-state lighting devices that convert electrical energy to light through electroluminescence [1]. A standard device configuration consists of a transparent anode, typically indium tin oxide, an organic semiconducting polymer film that is 10–1000 nm thick and a metallic cathode [2–5]. An applied voltage drives the injection of electrons through the cathode, and electron vacancies, or holes, through the anode into the organic thin film, where they diffuse and migrate, or drift, towards the oppositely charged electrode. When in close proximity, an electron and hole recombine to form an exciton, which radiatively decays to emit a photon.

The current in an organic device is limited by charge transport across the organic thin film [6] when the energy barrier to injection is lower than 0.3–0.4 eV [7], as opposed to being limited by the electrode–film contacts that provide the majority of the resistance in inorganic devices [8]. At the electrode–organic film interface, ‘space-charge layers’ exist where the charge carrier (electron or hole) density varies rapidly [9]. Quantifying the charge carrier distribution within these space-charge layers is crucial to understanding charge transport in OLEDs [9], as well as in other types of organic electronics such as light-emitting electrochemical cells [10] and organic solar cells [11].

The current and recombination rate are key parameters in analysing OLED performance. The simplest mathematical model that relates current, charge recombination and applied voltage is the drift–diffusion equations that account for diffusion, drift and recombination of electrons and holes [12–14]. The drift–diffusion equations are nonlinear coupled differential equations that cannot be solved exactly. However, when simplifying assumptions are made, it is possible to make analytical progress. For example, Mott & Gurney [12] considered diffusion-free single-carrier injection into a thin film with no intrinsic charge, for which the current is 1.1where is the applied voltage; is the width of the thin film; is the permittivity of the thin film; is the electric mobility, , where is the diffusivity of the charge carrier, is the magnitude of the charge of an electron, is Boltzmann's constant and is temperature. Note that all hatted variables (e.g. ) are dimensional. Equation (1.1) is known as the Mott–Gurney law [12].

For single-carrier injection, the Mott–Gurney law has been extended to include the effects of charge traps [15–19], uneven injection barriers [8,20] and intrinsic charge [17]. The latter is responsible for an ohmic regime () that precedes the space-charge-limited regime described by the Mott–Gurney law [12]. Mark & Lampert [17] add a term to the Mott–Gurney law that describes the linear increase in current with voltage owing to intrinsic charge. Murgatroyd [21] found that the current–voltage relation for single carrier, diffusion-free injection can be represented by a functional relationship between and , which is consistent with the Mott–Gurney law [12] and Mark & Lampert's ohmic current at low voltage [17].

For double-carrier transport, Parmenter & Ruppel [22] obtained a solution to the drift–diffusion equations for diffusion-free, double-carrier injection and recombination into trap-free insulators, where recombination is modelled as a bimolecular reaction according to Langevin kinetics [23]. Parmenter & Ruppel [22] found that the Mott–Gurney law holds if the single-carrier electric mobility is replaced with an effective mobility that is a function of the individual electron and hole mobilities, and a recombination mobility. In contrast to Parmenter & Ruppel [22], who neglected diffusion, Baron's [24] work included double-carrier diffusion, drift and recombination. In particular, Baron used Poisson's equation to account for deviations from electroneutrality in the film, yet did not include the contribution of the gradient in charge density to the total carrier flux across the thin film. Baron found that the Mott–Gurney scaling, , holds.

The increase in current owing to diffusion and double-carrier injection is examined in Torpey's [25] work. Torpey first addressed the diffusion-free case of double-carrier injection with recombination using a scaling analysis and confirmed that , in agreement with Parmenter & Ruppel [22]. Torpey then solved the drift–diffusion equations numerically and found that at small reservoir concentrations the current is limited by the inability of the contacts to provide sufficient electrons and holes. Torpey found a space-charge-limited current at large reservoir concentrations, where . Notably, at large reservoir concentrations, the current is much greater than the diffusion-free scaling (see fig. 11 in Torpey [25]).

Lastly, when diffusion and drift are included, but recombination is neglected, Neumann *et al*. [26] find at high voltage, and at low voltage compared with the thermal voltage, (which is 26 mV at 300 K). They assert that there is only one length scale in the system, the device thickness , to justify the scaling with in the current–voltage relation at all voltages. However, as we show below, when diffusion is included an additional length scale arises: the space-charge length.

The electrode–film boundary conditions imposed in the works discussed above feature ohmic contacts [22,24,26,27], or reservoir conditions that fix the concentrations at the electrodes [4,9,12,17,25]. The ohmic condition assumes that the electric field is zero at the electrodes, whereas reservoir conditions assume a fixed voltage-independent concentration. In addition to these boundary conditions, Walker *et al*. [28] and Malliaras & Scott [8] review boundary conditions that quantify the injection of charge due to thermionic emission, tunnelling and backflowing interface recombination, where injection is a function of the local electric field at the electrode–organic interface.

In this work, we derive a current–voltage relation for space-charge-limited transport in thin organic films owing to diffusion and drift of electrons and holes. Although this analysis does not account for carrier recombination, the results are subsequently used to derive a recombination–voltage relation for low recombination rate OLEDs.

As shown below, non-dimensionalization of the drift–diffusion equations reveals an important parameter governing charge transport in the thin film: the width of the space-charge layer,
1.2where is the charge density at the electrode. This is mathematically equivalent to the Debye screening length in electrochemical systems [29]. In the present case, however, the space-charge develops as a result of injection of charges into a film with no intrinsic charge, whereas the Debye length characterizes the extent of the space charge of mobile ions intrinsic to an electrolyte near a charged surface. Importantly, for space-charge-limited transport in OLEDs, the space-charge layer is often small in comparison with the width of the device, such that [4,9,30]. For instance, in Pinner *et al*.'s [4] experiments on OLED charge transport *ϵ*=0.048 for an ITO/PPV/Au device. In Torpey's [25] numerical simulation of small-molecule organic semiconductors, based on the material parameters used in the paper, *ϵ* ranges from 0.022 to 0.049. Similarly, deMello's [9] numerical work contains examples where *ϵ*=0.00098 for double-carrier injection into an OLED with ohmic contacts. In Peng *et al*. [30], numerical analysis of charge transport in OLEDs, *ϵ*=0.0012 for ohmic and injection limited cases.

When the space-charge layer is thin, *ϵ*≪1, numerical solutions of the drift–diffusion equations are challenging, because the equations are singular as [9]. This is manifested in the development of sharp gradients in charge carrier concentration: the charge accumulates in thin regions (space-charge layers) at each electrode and rapidly decays to a low bulk concentration. Asymptotic analysis, specifically singular perturbation methods, are ideal for such problems [31]; indeed, they exploit the fact that *ϵ*≪1 to obtain analytic, closed-form approximations that are asymptotic as . In this paper, we use these techniques to derive a current–voltage relation for OLEDs with double-carrier diffusion and drift.

This paper is structured as follows. In §2, we present the drift–diffusion equations that describe charge transport and recombination in an OLED. In §3 through §5, we apply asymptotic analysis to the drift–diffusion equations to derive a current–voltage relation for double-carrier injection into OLEDs in the thin space-charge-layer regime, *ϵ*≪1. In §6, we compare the current–voltage relation to the numerical solution of the drift–diffusion equations. In §7, we derive a recombination–voltage relation for kinetically limited OLEDs and compare with the numerical solution of the drift–diffusion equations with recombination. We present conclusions in §8.

## 2. Mathematical model

The drift–diffusion equations describe the transport of electrons and holes in an organic film, caused by diffusion down concentration gradients and drift in an electric field. The thickness (10–1000 nm) of an OLED is typically much smaller than its width and depth; thus, the device is treated as one-dimensional, and charge transport occurs normal to the electrodes only, designated as the direction. The -origin is located at the anode (figure 1). We assume that the OLED is operating at steady state. The net fluxes of each species are 2.1where the ± subscript refers to electrons (−) or holes (+), is the flux of the charge carrier, is the carrier mobility and is the electric potential. The first term in (2.1) accounts for diffusion, the second drift. Poisson's equation states that the variation in electric field across the OLED is proportional to the local charge density, 2.2Charge carrier conservation equations equate the change in flux across the film to the consumption of charge carriers due to recombination, 2.3where is the reaction rate constant for the recombination of electrons and holes, with units of length cubed per time, assuming Langevin kinetics [23]. In (2.3), we do not account for any electron or hole generation through exciton dissociation.

We assume that electrons are injected through the cathode and holes through the anode such that the number density of charge carriers at the electrode is finite and constant [25], and that the potential at the electrodes is equal to an applied potential. We assume that the concentration of a charge carrier is zero at its counter electrode [9,26]. These assumptions form the boundary conditions described in (2.12) and (2.13).

To non-dimensionalize the governing equations (2.1)–(2.3), the spatial coordinate, , is normalized by the device length, ; the electric potential by the thermal voltage; and the electron and hole densities by , the charge carrier density at the electrodes. The flux is normalized by . For the sake of simplicity, we set the magnitude of the charge density at the anode and at the cathode to be equal, and set the mobility of electrons equal to the mobility of holes. In addition, we neglect the electric field dependence of the mobility. Henceforth, all equations are in terms of dimensionless variables (unless stated otherwise) and appear without a caret accent. From (2.1), the flux is
2.4Poisson's equation now reads
2.5where *ϵ* is the dimensionless space-charge length, defined as
2.6From (2.3), the charge conservation equations are
2.7where the dimensionless reaction rate is known as the Damkhöler number in engineering literature [32], and is equal to the ratio of the diffusive time scale () to the recombination time scale (). When the reaction is slow in comparison with diffusion (*k*≪1), the recombination is kinetically limited. If the inverse is true, then recombination is limited by charge transport. Here, we assume that the system is kinetically limited, so the reaction term can be neglected to a first approximation (*k*=0) in (2.7) yielding
2.8

It is convenient to write the drift–diffusion equations (2.4)–(2.8) in terms of the mean charge carrier concentration, , half the charge density, and the current, *J*=*j*_{+}−*j*_{−}. Thus, Poisson's equation becomes
2.9The net flux, *j*_{+}+*j*_{−}, is zero across the film, because the flux of electrons and holes are equal in magnitude but opposite in direction, which implies
2.10An integration of (2.8) shows that the current *J* satisfies
2.11

The aforementioned boundary conditions, in non-dimensional form, are
2.12and
2.13where *V* is the applied potential (normalized by the thermal voltage, ) minus the normalized built-in potential drop owing to the difference in the work functions of the anode and cathode [33]. The applied potential *V* is equal to the integral of the electric field *E* across the device,
2.14where
2.15A schematic of the device configuration is shown in figure 1. In the proceeding analysis, we solve for the electric field, hole and electron density, and electric potential in the anodic half () of the OLED. The electric field is symmetric around , whereas the electric potential is antisymmetric. The electron density is symmetric to the hole density, so asymptotic expressions for the electron density in the cathodic region () can be derived from the expressions for the hole density in the anodic region by replacing *x* with 1−*x*, and vice versa.

In the absence of recombination, the problem presented here is mathematically similar to ion transport across a permeable membrane at the interface between two reservoirs of fixed electrolyte concentrations [34–36], relevant to ionic channels in biological systems. Note, however, that in [34–36] it is assumed that the concentrations of both species (cation and anion) are non-zero in each reservoir. In the case of OLEDs, the appropriate boundary conditions set the concentration of electrons (holes) to zero at the anode (cathode). Another related electrodiffusion problem is that of current passage across a cation-selective surface, which is relevant to electrodialysis at over limiting currents in thin-gap cells [37]. In that problem, the anions do not react at the electrodes; hence, their concentration within the cell is subject to an integral constraint. For an OLED, there are no such constraints on the concentrations of electrons and holes in the semiconductor film.

## 3. Master equation for the electric field

Equations (2.9)–(2.11) can be combined into a single master equation in terms of the electric field *E*. First, (2.9) is substituted into (2.10), yielding
3.1Equation (3.1) is readily integrated, resulting in
3.2where the constant of integration is
3.3where *E*_{⋆} and *c*_{⋆} are the electric field and mean concentration, respectively, at the midpoint of the device. These two quantities are not known *a priori*, but must be determined as part of the asymptotic analysis. From (3.2) and (3.3), the carrier concentration can be written as
3.4Poisson's equation (2.9) is inserted into (2.11) resulting in
3.5Equation (3.4) is combined with (3.5) to yield the master equation for the electric field across the OLED,
3.6Similar master equations have been derived to analyse charge transport in liquid-state electrochemical systems [36,38] and in OLEDs without diffusion [25].

## 4. Asymptotic analysis

In the operation of an OLED, a potential difference between the anode and the cathode drives a current across the organic thin film. The applied voltage must be high enough to overcome the energy barriers for the injection of charge carriers. Typically, the potential difference is much greater than the thermal voltage, *V* ≫1 (or mV) [2]. Therefore, we prescribe the voltage
4.1where *V* _{⋆} is an *O*(1) constant and *μ*(*ϵ*)≫1. The bulk field, *E*_{B}, arises from the *O*(*μ*) drop in potential across the *O*(1)-width bulk region. Owing to symmetry of the field about , the integral (2.14) can be written as
4.2where we have used , and *ϕ*(0)=*V*/2=*μV* _{⋆}/2. Hence, the bulk field is
4.3which is spatially uniform to leading order. Therefore, the electric field at the midpoint is *E*_{⋆}=*μ*(*ϵ*)*V* _{⋆} to leading order. The majority of the injected charge carriers in the device are concentrated at the space-charge layers near each electrode, so the concentration in the electroneutral bulk is expected to be asymptotically small in *ϵ*,
4.4where *δ*(*ϵ*)≪1 is an *a priori* unknown function and *c*_{a} is an *O*(1) coefficient. Thus, *c*_{⋆}=*δ*(*ϵ*)*c*_{a} to leading order. In the bulk, where *x*∼*O*(1), one can simply set *ϵ*=0 in the master equation (3.6) to derive the leading-order current *J* as
4.5The charge density is zero in the electroneutral bulk, to leading order. However, at the anode–thin-film interface (*x*=0) the charge density and at the cathode (*x*=1), according to (2.12) and (2.13). The bulk solution (4.3)–(4.5) does not satisfy these boundary conditions. This indicates the existence of an asymptotically small region near each electrode, a space-charge layer, where electroneutrality is violated. We derive an asymptotic approximation for the electric field in the space-charge layer next.

### (a) Space-charge layer

Owing to symmetry, we focus on the hole-rich space-charge layer near the anode (*x*=0). In this region, the concentration of charge carriers varies rapidly; hence, we define a stretched ‘inner’ coordinate , such that as . The electric field in the space-charge layer is expected to be *O*(1/*ϵ*), resulting in an *O*(1) potential drop across the space-charge-layer (the majority of the *O*(*μ*) applied potential drops across the bulk of the OLED). We define , where . Applying these scalings to the master equation (3.6) yields
4.6Recall that the midpoint concentration *c*_{⋆} (4.4) is small, *δ*≪1, and the midpoint field *E*_{⋆} (4.3) is *O*(*μ*). Thus, to ensure that the space-charge layer is in quasi-equilibrium to a first approximation, we choose *μ*≪1/*ϵ*, so that terms proportional to the midpoint field *E*_{⋆} and current *J* (∼*μδ*) in (4.6) do not enter the leading-order balance. That leading *O*(1/*ϵ*) balance is therefore
4.7with the boundary conditions
4.8derived from (2.9), (2.12) and (3.6). The solution to (4.7) and (4.8) is
4.9which represents the leading-order solution for the electric field in the space-charge layer. This expression was also found by Chu & Bazant [39] for the electric field in the Debye layer of an electrochemical cell at the diffusion-limited current.

The decay to zero of the *O*(1/*ϵ*) space-charge field as is consistent with the prescribed *O*(*μ*) bulk field, where *μ*≪1/*ϵ*. We look to (4.6) for the scaling of the next term of the space-charge layer field. After the terms included in (4.7), the next largest term in (4.6) is from (4.4), because *c*_{a} and are *O*(1), and *μ*≪1/*ϵ*. Thus, the expansion of electric field in the space-charge layer is
4.10where is the previously calculated leading-order term (4.9). An equation for the next, *O*(*δ*/*ϵ*), term of the electric field in the space-charge layer, , is derived by inserting (4.10) into (4.6). Because *μ*≪1/*ϵ*, the current *J* does not enter the *O*(*δ*/*ϵ*) balance. The term is to leading order from (4.4). The term containing the midpoint field, is *O*(*ϵμ*^{2}) to leading order in . When *μ* is restricted to *μ*≪*δ*^{1/2}/*ϵ*, this term does not enter into the *O*(*δ*/*ϵ*) balance. With this restriction, the *O*(*δ*/*ϵ*) field is governed by
4.11with the boundary conditions
4.12Substituting the result for (4.9) into (4.11) and solving the resulting equation yields
4.13The *O*(*δ*/*ϵ*) field (4.13) diverges as and hence clearly does not match the bulk, indicating that a transition, or intermediate, region exists between the electroneutral bulk and the space-charge layer.

### (b) Intermediate layer

The device length and the width of the space-charge layer are clear choices for the characteristic length of the bulk, *x*∼*O*(1), and the space-charge layer, *x*∼*O*(*ϵ*), respectively; however, there is no obvious length scale for the intermediate layer. Through a scaling analysis, the intermediate layer field and width are found to scale as
4.14where and are *O*(1). The width of the intermediate layer is *ϵ*/*δ*^{1/2}, and the magnitude of the field within it is *δ*^{1/2}/*ϵ*, which fit the criteria *ϵ*≪*ϵ*/*δ*^{1/2}≪1 and *μ*≪*δ*^{1/2}/*ϵ*≪1/*ϵ*. Therefore, the leading-order equation that results in the intermediate layer from (3.6) is
4.15A solution to (4.15) that decays to zero as is [40]
4.16

The behaviour of the asymptotic expansion for the intermediate field must match that of the space-charge field as and the bulk field as . As , ; as , the leading-order space-charge field (4.9), rewritten in terms of and , approaches . Therefore, the behaviour of the intermediate field as matches that of the space-charge field. Because we require *μ*≪*δ*^{1/2}/*ϵ*, the decay of (4.16) at large is consistent with the *O*(*μ*) bulk field (4.3). To determine a relation between *μ*, *δ* and *ϵ*, we need a second term for the field in the intermediate layer that scales by *μ* to match to the leading-order bulk electric field *E*_{B}=*μV* _{⋆}. The expansion for the field in the intermediate region is thus written as
4.17where the leading-order term is given by (4.16). From (3.6), the equation for is
4.18where *c*_{b} is the next order of the expansion of the midpoint concentration that is
4.19which is added to account for a possible higher-order bulk concentration contribution to . An exact solution to (4.18) cannot be obtained. However, only the behaviour of as and as is required for matching with the bulk and the space-charge layer, respectively. As it is clear that , because decays exponentially, thereby matching the bulk field, but the behaviour as needs to be examined carefully.

A power series expansion around reveals two homogeneous solutions to (4.18):
4.20and
4.21where *a*_{0} and *b*_{0} are constants, as well as a particular solution,
4.22The series approximation of the *O*(*μ*) correction to the intermediate field as is thus
4.23We now proceed to match the intermediate field, (4.16) and (4.23), to the electric field in the space-charge layer, (4.9) and (4.13), according to van Dyke's matching procedure [31], in order to obtain *a*_{0} and *δ*. This is detailed in appendix A. The results of the matching procedure are
4.24However, we have not yet found an expression for *c*_{a}, which can be determined by matching only to higher-order space-charge-field terms.

### (c) Higher-order space-charge-field terms

To determine an expression for *c*_{a}, where *c*_{B}∼*δc*_{a} is the leading-order bulk concentration, we need to determine the next two terms in the space-charge layer field, which include the contribution of current. The order of these terms in the space-charge layer expansion can be determined from the higher-order terms in the correction to the intermediate field (4.23), as . Equation (4.23) implies the expansion
4.25Substituting (4.25) into (4.6), the differential equation for is
4.26subject to the boundary conditions
4.27This differential equation has the same form as (4.11); the solution has the same form as (4.13), except that *c*_{a} is replaced by *c*_{b}. Thus,
4.28This term matches to the term in (4.23), however, this does not provide the value of *c*_{a}. Another term of the expansion of the space-charge layer field is required here to determine *c*_{a}. For the next term, , we first extend the expansion of the midpoint concentration *c*_{⋆} by an additional term
4.29where we have applied *μ*=*δ*/*ϵ* from (4.24). Equation for the fourth term in the space-charge layer electric field expansion is
4.30subject to
4.31Equations (4.30) and (4.31) can be solved in closed form; however, we are interested only in the behaviour as . A power series expansion of the solution to (4.30) and (4.31) as yields
4.32Next, we match the first four terms in the expansion for the space-charge field to the first two terms in the intermediate solution. This matching procedure, in contrast to the last, matches the terms in each expansion that are larger than *O*(*δ*^{2}/*ϵ*) in the space-charge field expansion and *O*(*δ*/*ϵ*) in the intermediate field expansion according to van Dyke's rule [31]. This procedure, detailed in appendix A, yields
4.33When (4.1) and (4.24) are inserted into (4.33), the bulk concentration is recovered to leading order as
4.34which is indeed small as *V* =*μV* _{⋆}, where *μ*≪1/*ϵ*.

## 5. Current–voltage relation

From (4.3), (4.5) and (4.34), the leading-order current is
5.1The first correction to (5.1) can be found through a higher-order analysis of the integral constraint (2.14). The details of the analysis are given in appendix B, where it is shown that the constraint yields a logarithmic correction to the leading-order bulk electric field,
5.2Substituting (4.34) and (5.2) into (4.5) yields the improved current–voltage relation,
5.3From (5.2) and (5.3), for the analysis to be valid the applied voltage must be sufficiently large such that . An upper bound on the applied voltage, *μ*≪*δ*^{1/2}/*ϵ*, is necessary to calculate the first correction to the space-charge field (4.23). The current–voltage relation (5.3) is therefore valid over the range
5.4where *μ* is prescribed by the magnitude of the applied voltage *V* . The current (5.3), in dimensional form, reads
5.5where is given by (1.2). To leading order, the current exhibits a quadratic dependence on the applied voltage and the reciprocal of the film width , , in contrast to the Mott & Gurney law for diffusion-free single carrier injection, where .

## 6. Comparison between asymptotic analysis and numerical solution of the drift–diffusion equations

Our analysis has furnished asymptotic approximations for the field and carrier densities across the OLED. To validate our analysis, we solved the drift–diffusion equations (2.4), (2.5) and (2.8) numerically using the Matlab BVP4C solver. The asymptotic results for the electric field are compared against the numerical results for *ϵ*=0.001 (figure 2), showing good agreement.

The density of holes *n*_{+} and electrons *n*_{−} in the anodic region is easily calculated from the asymptotic expressions for the electric field. In figure 3, the hole and electron densities in the anodic region are plotted on a semilog axis. The solution for the hole density in the space-charge layer includes the first correction to the density, calculated from (4.13). The asymptotic expansion for the electron density in the intermediate region decays to zero as (figure 3), as expected; in fact, from (4.9), (4.13) and (4.28), it can be shown that the electron density is zero through *O*(*δ*^{3/2}). The discrepancy between the asymptotic bulk electron and hole densities and the numerics can be attributed to the fact that only the leading-order term for the bulk concentration (4.34) was calculated. This error is seen in both the electron and hole density asymptotic expansions.

The steep gradient in hole density in figure 3 results in a diffusive current that must be balanced by a negative electric field to conserve charge in (2.8). From the definition of the space-charge width (2.6), . Hence, as the charge density in the electrode approaches infinity, as in the case of ohmic contacts without diffusion, . A plot of the potential at several values of *ϵ* is useful (figure 4) to elucidate the increase in current owing to diffusion. The leading-order potential *ϕ* is calculated from (2.15) and the electric field across the space-charge layer (4.9), the intermediate layer (4.16) and the bulk (5.2).

As , a maximum in potential develops, resulting in ‘virtual ohmic contacts’ [25] near the electrodes. The maximum results from the negative electric field near the anode that develops to balance diffusion in the charge conservation equation (2.8). The virtual contacts are located at the point where the electric field is zero. Between the virtual contacts near the anode and the cathodes, the voltage drops linearly. The potential at the virtual contact increases as *ϵ* decreases, and is larger than the applied potential at the electrode. Hence, the voltage drop across the bulk is larger than the naive estimate , resulting in a higher current. This is reflected in the current–voltage relation (5.5), where . If diffusion is neglected this increase in current due to an interior ‘virtual contact’ across the OLED is not accounted for.

The current–voltage relation (5.3) is compared with the numerical results in figure 5 at three values of *ϵ*. As *ϵ* decreases, the accuracy of the asymptotic expression increases. The current–voltage relation (5.3) is fundamentally different from the Mott–Gurney law for diffusion-free injection into insulators. The major difference is the additional characteristic length scale, the space-charge length , that results from including diffusion: instead of , we find that to leading order. Because the space-charge layer is much thinner than the film, , this predicts a large increase in the current that passes through the film for double-carrier injection with diffusion (figure 5). This can be interpreted in terms of the virtual ohmic contacts as discussed above.

The increase in current is consistent with experimental results for double-carrier injection into an ITO/MEH-PPV/Ca OLED performed by Parker [41]. The energy barrier to injection across the MEH-PPV/Ca interface is approximately 0.1 eV, which indicates a space-charge-limited OLED [7]. Specifically, Parker reports current–voltage curves for OLEDs for varying widths of the polymer thin film, . When the macroscopic electric field across the film, , is plotted against the current, the curves for the various widths collapse onto a single current–voltage curve, indicating that the current–voltage relation should be a function of the ratio , as seen in (5.5), where , as opposed to for diffusion-free transport. In terms of the charge density at the electrode, , the current increases with the reservoir density, as expected. When the charge density in the electrode is small, the current is small because it is limited by the electrons and holes available in the electrode for transfer to the thin film.

## 7. The first effects of carrier recombination

Thus far, we have set the recombination rate constant *k* equal to zero, thereby neglecting recombination of carriers. The results reported in §6 for the electric field, hole and electron densities, and electric potential, can be viewed as the leading-order terms in an asymptotic expansion in terms of the recombination rate. That is, the leading-order local recombination rate is
7.1where *n*_{+}(*x*;*ϵ*) and *n*_{−}(*x*;*ϵ*) are the hole and electron profiles calculated for *k*=0. In the space-charge layer adjacent to the anode, the electron density is zero to *O*(*δ*^{3/2}); thus, the local recombination rate in the space-charge layer is zero through *kδ*^{3/2}. In the intermediate region, the leading-order recombination rate is equal to the leading-order bulk recombination rate. The total recombination rate across the OLED is the integral of the local recombination rate, . Because the space-charge layers are thin, the bulk recombination rate *r*∼4*kϵ*^{2}*V* ^{2}, from (4.34), is the dominant contribution to the total recombination rate. The leading-order asymptotic expression for the total recombination rate,
7.2is compared with numerical results in figure 6. In figure 6, the numerics solve the full drift–diffusion equations, including recombination, at *ϵ*=0.001 and *V* =50, for a variety of recombination rate constants, *k*. The asymptotic solution matches well to the numerics through *O*(1) values of *k*, indicating that the simple expression *R*∼4*kϵ*^{2}*V* ^{2} may be applied to OLEDs to approximate the total recombination rate for low-to-moderate recombination rate constants.

## 8. Conclusions

We have quantified double-carrier drift, diffusion and recombination in OLEDs via asymptotic analysis of the drift–diffusion equations. For space-charge-limited OLEDs, the carrier densities vary rapidly near the electrodes. We have shown this variation in carrier densities occurs on the length-scale of the space-charge width , which is much smaller than the device thickness. Thus, the ratio is a small parameter exploited in our analysis to yield asymptotic approximations to the current–voltage relation and the electric field, carrier densities and electric potential across the OLED. Chiefly, we found that the leading-order current across the OLED is , in contrast to diffusion-free single-carrier injection, where [12]. The difference is due to the characteristic length scale, the space-charge width , that emerges when the drift–diffusion equations are non-dimensionalized. This space-charge width is much thinner than the width of the device, so we predict a higher current than a diffusion-free analysis would, which can be interpreted as an increased potential drop between virtual ohmic contacts that result from diffusion. The scaling for the current derived here, , is consistent with experimental data for ITO/MEH-PPV/Ca OLEDs by Parker [41], which shows that the current scales as .

We assumed that the charge carrier densities at the electrode were equal, and the electrons and holes have the same mobility in the organic film. In addition, we assumed that the charge carrier mobility does not depend upon the electric field. These assumptions can be relaxed; we plan to do so in future work. For example, the mobility can be adjusted to account for the electric field through the functional form , where and are material parameters [7].

Lastly, we analysed kinetically limited OLEDs with small recombination rates (*k*<1). We derived an approximate expression for the recombination rate across the OLED that compares well with numerics for low-to-moderate reaction rate constants *k*. A solution extended to apply at larger *k* is desirable to accurately predict the recombination in an OLED beyond the kinetically limited regime.

## Acknowledgements

S.E.F. acknowledges support by the National Science Foundation Graduate Research Fellowship under grant no. 0946825. O.S. thanks the United States– Israel Binational Science Foundation (BSF) for awarding him the Professor Rahamimoff travel grant, thereby facilitating this collaboration.

## Appendix A. Matching

To determine the value of *a*_{0} in (4.23), we match the field in the space-charge layer to that of the intermediate layer in the domain of overlap, and . As , from (4.9) and (4.13) we have
A1The expansion of the leading-order intermediate field (4.16) as is
A2whereas the next term has the expansion (4.23). Next, we rewrite the space-charge layer expansions (A1) in terms of the intermediate coordinate , collect terms, and then rewrite the intermediate expansion in terms of space-charge layer coordinate . This matching procedure is discussed by van Dyke [31]. The space-charge layer expansions (A1) in terms of , where , are
A3Recall, *δ*≪1. The expansion of the intermediate field, (4.23) and (A2), in terms of the inner variable is
A4For (A3) and (A4) to be equal
A5

To determine *c*_{a} we match the four terms in the space-charge layer expansion (4.9), (4.13), (4.28) and (4.32) to the two-term intermediate expansion (4.16) and (4.23). The series approximations for and as and as are (A3) and (A4), respectively. The *O*(*δ*^{3/2}/*ϵ*) correction to the space-charge field, , as is
A6whereas the *O*(*δ*^{2}/*ϵ*) correction, , is approximated as (4.32) as . The first four terms in the expansion for the space-charge layer field (, , and ) are rewritten as functions of the intermediate spatial variable, , through the relation, , which gives
A7The power series expansion of the first two terms representing the intermediate field as is given by (A2) and (4.23). These terms are rewritten in terms of the space-charge layer spatial coordinate as
A8After rewriting (A7) and (A8), it is clear that the following matching conditions result:
A9Both these conditions yield the same expression for *c*_{a}, namely
A10

## Appendix B. Integral constraint

An integral constraint (4.2) relates the applied potential to the integral of the electric field across the thin film. The integral (4.2) is broken up into three regions, corresponding to the electric field profiles in the space charge layer, intermediate layer and in the bulk. Thus,
B1In the space charge layer, the lower bound is zero at the anode, whereas the upper bound is a regularization variable *η* normalized by the space-charge layer width *ϵ*, where *ϵ*≪*η*≪*ϵ*/*δ*^{1/2}. In the intermediate layer, the bounds are the variables *η* and *γ* normalized by the intermediate layer width *ϵ*/*δ*^{1/2}, where *ϵ*/*δ*^{1/2}≪*γ*≪1. The purpose of *η* and *γ* is to avoid diverging contributions from the space-charge layer and intermediate integrals as , and respectively. The final result cannot depend on *η* or *γ*.

The integral of the space-charge field is
B2where the error originates from the integral of the next term, , across the space charge layer. The integral of the leading order field (4.9) is
B3whereas the integral of the first correction (4.13) is
B4In the intermediate layer, the integral of the leading-order field (4.16) is
B5Finally, the integral of the bulk field, whose lower limit *γ* is replaced by 0, because the bulk field is a constant to leading order, is
B6The contributions (B3)–(B6) are summed, resulting in
B7Thus, (B1) is asymptotic to (B7) to *O*(*μ*); however, there is an mismatch. Therefore, the bulk field is adjusted to . Note that this adjustment of *E*_{B} does not affect any of the preceding analysis.

The new expression for the bulk electric field, after substituting for *μV* _{⋆}, *δ* and *c*_{a} according to (4.1), (4.24), and (4.33), respectively, is thus
B8

- Received April 25, 2013.
- Accepted July 2, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.